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Mathematics > Metric Geometry

Title:Entropy-based Bounds on Dimension Reduction in L_1

Abstract: We show that for every large enough integer $N$, there exists an $N$-point
subset of $L_1$ such that for every $D>1$, embedding it into $\ell_1^d$ with
distortion $D$ requires dimension $d$ at least $N^{Ω(1/D^2)}$, and that
for every $\eps>0$ and large enough integer $N$, there exists an $N$-point
subset of $L_1$ such that embedding it into $\ell_1^d$ with distortion $1+\eps$
requires dimension $d$ at least $N^{1-O(1/\log(1/\eps))}$. These results were
previously proven by Brinkman and Charikar [JACM, 2005] and by Andoni,
Charikar, Neiman, and Nguyen [FOCS 2011]. We provide an alternative and
arguably more intuitive proof based on an entropy argument.